Numerical method for solving matrix coefficient elliptic equation with sharp-edged interfaces

Hou, Songming; Wang, Wei; Wang, Liqun

doi:10.1016/j.jcp.2010.06.005

Cited by 77 publications

(100 citation statements)

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“…In [26][27][28][29][30][31][32], jump conditions and are known and are used to derive the linear combination of the values of the points across the interface Γ. However, in this paper, we need to construct and for each interface cell Δ ∈ Λ 2 from the points with known value across the interface Γ and − on Ω − and then calculate the linear integral…”

Section: Construction Of Jump Conditions and Weak Formulationmentioning

confidence: 99%

“…The idea of applying interface problem solvers into noninterface problems is completely new. The nontraditional finite element method is originally developed in [26][27][28][29][30][31][32] for solving elliptic or elasticity equations with sharp-edged interfaces. This method uses non-body-fitting Cartesian grids and uses different basis for the solution and test function; its linear system is independent of jump condition.…”

Section: Introductionmentioning

confidence: 99%

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Numerical Method for Solving Matrix Coefficient Elliptic Equation on Irregular Domains with Sharp-Edged Boundaries

Wang

Shi

2013

Advances in Numerical Analysis

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We present a new second-order accurate numerical method for solving matrix coefficient elliptic equation on irregular domains with sharp-edged boundaries. Nontraditional finite element method with non-body-fitting grids is implemented on a fictitious domain in which the irregular domains are embedded. First we set the function and coefficient in the fictitious part, and the nonsmooth boundary is then treated as an interface. The emphasis is on the construction of jump conditions on the interface; a special position for the ghost point is chosen so that the method is more accurate. The test function basis is chosen to be the standard finite element basis independent of the interface, and the solution basis is chosen to be piecewise linear satisfying the jump conditions across the interface. This is an efficient method for dealing with elliptic equations in irregular domains with non-smooth boundaries, and it is able to treat the general case of matrix coefficient. The complexity and computational expense in mesh generation is highly decreased, especially for moving boundaries, while robustness, efficiency, and accuracy are promised. Extensive numerical experiments indicate that this method is second-order accurate in the ∞ norm in both two and three dimensions and numerically very stable.

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Section: Construction Of Jump Conditions and Weak Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Numerical Method for Solving Matrix Coefficient Elliptic Equation on Irregular Domains with Sharp-Edged Boundaries

Wang

Shi

2013

Advances in Numerical Analysis

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“…These methods were extended to the case of Crouzeix-Raviart P 1 nonconforming finite element method [12] by Kwak et al [18], and to the problems with nonzero jumps in [7]. Some related works on interface problems can be found in [5], [16], [17], [19], [22], [23], [26].…”

Section: Introductionmentioning

confidence: 99%

A Modified $P_1$ -- Immersed Finite Element Method

Kwak¹,

Lee²

2015

Int. J. of Pure and Appl. Math.

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In recent years, the immersed finite element methods (IFEM) introduced in [20], [21] to solve elliptic problems having an interface in the domain due to the discontinuity of coefficients are getting more attentions of researchers because of their simplicity and efficiency. Unlike the conventional finite element methods, the IFEM allows the interface to cut through the interior of the element, yet after the basis functions are altered so that they satisfy the flux jump conditions, it seems to show a reasonable order of convergence.In this paper, we propose an improved version of the P 1 based IFEM by adding the line integral of flux terms on each element. This technique resembles the discontinuous Galerkin (DG) method, however, our method has much less degrees of freedom than the DG methods since we use the same number of unknowns as the conventional P 1 finite element method.We prove H 1 and L 2 error estimates which are optimal both in order and regularity. Numerical experiments were carried out for several examples, which show the robustness of our scheme.

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“…In [13][14][15][16], we developed a nontraditional FEM with non-body-fitting grids for elliptic equations with smooth/sharpedged interfaces and ½u -0 and ½b @u @n -0. It can deliver (almost) second-order accuracy in L 1 -norm for both smooth and sharp-edged interfaces [14][15][16].…”

Section: Introductionmentioning

confidence: 99%

A simple method for matrix-valued coefficient elliptic equations with sharp-edged interfaces

Wang

Shi

2014

Applied Mathematics and Computation

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Numerical method for solving matrix coefficient elliptic equation with sharp-edged interfaces

Cited by 77 publications

References 20 publications

Numerical Method for Solving Matrix Coefficient Elliptic Equation on Irregular Domains with Sharp-Edged Boundaries

Numerical Method for Solving Matrix Coefficient Elliptic Equation on Irregular Domains with Sharp-Edged Boundaries

A Modified $P_1$ -- Immersed Finite Element Method

A simple method for matrix-valued coefficient elliptic equations with sharp-edged interfaces

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